TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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Bernstein polynomials converge uniformly to every continuous function on $[0,1]$

Statement

If $f:[0,1]\to\mathbb R$ is continuous, then $B_n(f)\to f$ uniformly on $[0,1]$ .

Facts & Assumptions

Given: A continuous function $f:[0,1]\to\mathbb R$ and $\varepsilon>0$ .

[L1]

A continuous function on a compact metric space is uniformly continuous (Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[L2]

The Bernstein basis has the zeroth and centred second moment identities (The zeroth, first, and second centred moments of the Bernstein basis).

[L3]

For each positive real $\eta$ there is a natural $N\ge1$ with $1/N<\eta$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

Proof

technique · direct

1.1

Choose $\delta>0$ such that $|f(s)-f(x)|<\varepsilon/2$ whenever $|s-x|<\delta$ , and choose $M$ with $|f|\le M$ .

L1choose

1.2

On the far part, $(k/n-x)^2\ge\delta^2$ ; hence its total basis weight is at most $x(1-x)/(n\delta^2)\le1/(4n\delta^2)$ by [L2].

L2algebra

2.1

Split the Bernstein sum into $|k/n-x|<\delta$ and its complement. The near part is at most $\varepsilon/2$ by the zeroth identity.

step 1.1L2algebra

2.2

Choose $n$ so large that $2M/(4n\delta^2)<\varepsilon/2$ . The far part is then below $\varepsilon/2$ , uniformly in $x$ .

step 1.2L3algebra

3.1

The near and far estimates give $|B_n(f)(x)-f(x)|<\varepsilon$ for every $x$ and all sufficiently large $n$ .

step 2.1step 2.2∎

Depends on

Used by

Polynomials are uniformly dense in C([0,1],ℝ) Corollary

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 58 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Bernstein polynomial (Encyclopedia of Mathematics) (standard reference, not scraped)

Bernstein polynomials converge uniformly to every continuous function on [0,1][0,1][0,1]